Skew-Hermitian matrix

In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A^* is its negative. That is, if it satisfies the relation:

$A^{+}=-A\;$

or in component form, if A = (a_i,j):

$a_{i,j}=-{\overline {a_{j,i}}}$

for all i and j.

Skew-Hermitian matrices can be understood as the matrix analogue of the pure imaginary numbers.

Examples

For example, the following matrix is skew-Hermitian:

${\begin{bmatrix}i&2+i\\-2+i&3i\end{bmatrix}}$

Properties

• The eigenvalues of a skew-Hermitian matrix are all purely imaginary. Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.
• All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, ie. on the imaginary axis. This definition includes the number 0i.
• If A is skew-Hermitian, then iA is Hermitian
• If A, B are skew-Hermitian, then aA + bB is skew-Hermitian for all real scalars a, b.
• If A is skew-Hermitian, then A^2k is Hermitian for all positive integers k.
• If A is skew-Hermitian, then A raised to an odd power is skew-Hermitian.
• If A is skew-Hermitian, then e^A is unitary.
• The difference of a matrix and its conjugate transpose ($C-C^{*}$) is skew-Hermitian.
• An arbitrary (square) matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:

$C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}(C+C^{*})\quad {\mbox{and}}\quad B={\frac {1}{2}}(C-C^{*}).$

• The space of skew-Hermitian matrices forms the Lie algebra u(n) of the Lie group U(n).

See also

• Hermitian matrix
• Normal matrix
• Skew-symmetric matrix
• Unitary matrix